Let ƒ be a function whose domain is the set X, and whose range is the set Y.

Let ( M, g ) be a Riemannian manifold and ƒ: M < sup > m </ sup > → R < sup > n </ sup > a short C < sup >∞</ sup >- embedding ( or immersion ) into Euclidean space R < sup > n </ sup >, where n ≥ m + 1.

Let ƒ and g be functions.

Let ƒ and g be two functions satisfying the above hypothesis that ƒ is continuous on I and is continuous on the closed interval.

Let k be an algebraically closed field and let A < sup > n </ sup > be an affine n-space over k. The polynomials ƒ in the ring k ..., x < sub > n </ sub > can be viewed as k-valued functions on A < sup > n </ sup > by evaluating ƒ at the points in A < sup > n </ sup >, i. e. by choosing values in k for each x < sub > i </ sub >.

Let k be the field of complex numbers C. Let A < sup > 2 </ sup > be a two dimensional affine space over C. The polynomials f in the ring Cy can be viewed as complex valued functions on A < sup > 2 </ sup > by evaluating ƒ at the points in A < sup > 2 </ sup >.

Let ƒ be measurable, E (| ƒ |) < ∞, and T be a measure-preserving map.

Let ƒ be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided by the angular measurement of the circle.

Let M and N be differentiable manifolds and ƒ: M → N be a differentiable map between them.

where stands for all partial derivatives of ƒ up to order k. Let us exchange every variable in for new independent variables

Let Ω be a connected open set in the complex plane C and ƒ: Ω → C a holomorphic function.

Let γ be a simple rectifiable loop in X around P. The ramification index of ƒ at P is

Let ƒ: X → Y be a morphism of algebraic curves.

Let Q = ƒ ( P ) and let t be a local uniformizing parameter at P ; that is, t is a regular function defined in a neighborhood of Q with t ( Q ) = 0 whose differential is nonzero.

Let ƒ ( x ) = ƒ ( x, y ) be a continuous function vanishing outside some large disc in the Euclidean plane R < sup > 2 </ sup >.

Let K be the sheaf of meromorphic functions and O the sheaf of holomorphic functions on M. A global section ƒ of K passes to a global section φ ( ƒ ) of the quotient sheaf K / O.

Let V and W be handlebodies of genus g, and let ƒ be an orientation reversing homeomorphism from the boundary of V to the boundary of W. By gluing V to W along ƒ we obtain the compact oriented 3-manifold

Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

Let g be a smooth function on N vanishing at f ( x ).

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).

Let X be a topological space, and let x < sub > 0 </ sub > be a point of X.

Let x < sub > 0 </ sub >, ...., x < sub > N-1 </ sub > be complex numbers.

Let now x ' and y ' be tuples of previously unused variables of the same length as x and y respectively, and let Q be a previously unused relation symbol which takes as many arguments as the sum of lengths of x and y ; we consider the formula

Let the open enemy to it be regarded as a Pandora with her box opened ; ;

Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.

`` Let him be now ''!!

Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let Af be the null space of Af.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.

Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.

Let this be denoted by Af.

Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.

Let not your heart be troubled, neither let it be afraid ''.

The same God who called this world into being when He said: `` Let there be light ''!!

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.

Let her out, let her out -- that would be the solution, wouldn't it??